This paper considers the problem of computing the real convex hull of a finite set of ndimensional integer vectors. The starting point is a finite-automaton representation of the initial set of vectors. The proposed method consists in computing a sequence of automata representing approximations of the convex hull and using extrapolation techniques to compute the limit of this sequence. The conv...

In this paper we want to introduce an algorithm for the creation of polyhedral approximations for objects represented as strongly connected sets of voxels in three-dimensional binary images. The algorithm generates the convex hull of a given object and modifies the hull afterwards by recursive repetitions of generating convex hulls of subsets of the given voxel set or subsets of the background ...

In this paper, we arithmetically describe the convex hull of a digital straight segment by three recurrence relations. This characterization gives new insights into the combinatorial structure of digital straight segments of arbitrary length and intercept. It also leads to two on-line algorithms that computes a part of the convex hull of a given digital straight segment. They both run in consta...

The dynamic maintenance of the convex hull of a set of points in the plane is one of the most important problems in computational geometry. We present a data structure supporting point insertions in amortized O(log n · log log logn) time, point deletions in amortized O(log n · log logn) time, and various queries about the convex hull in optimal O(log n) worst-case time. The data structure requi...

We consider the Minkowski sum of subsets of integer lattice, each of which is a set of integer points of a face of an extended submodular [Kashiwabara–Takabatake, Discrete Appl. Math. 131 (2003) 433] integer polyhedron supported by a common positive vector. We show a sufficient condition for the sum to contain all the integer points of its convex hull and a sufficient condition for the sum to i...

In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation o...

In this paper we present several results on the expected complexity of a convex hull of n points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of n points, chosen uniformly and independently from a disk is O(n1/3), and O(k log n) for the case a convex polygon with k sides. Those results are well known (see [RS63, Ray7...

For a connected graph G with at least two vertices and S a subset of vertices, the convex hull [S]G is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V (G) with [S]G = V (G). Upper bound for the hull number of strong product G⊠H of two graphs G and H is obtainted. Improved upper bounds are obtained for some class of strong product gr...

a r t i c l e i n f o a b s t r a c t Keywords: Graph algorithms Simple-path convexity Simple-path convex hull Totally balanced hypergraphs In a connected hypergraph a vertex set X is simple-path convex (sp-convex, for short) if either |X| 1 or X contains every vertex on every simple path between two vertices in X (Faber and Jamison, 1986 [7]), and the sp-convex hull of a vertex set X is the mi...

This theorem is similar to the well known result that any point in the convex hull of a set E in an w-dimensional euclidean space lies in the convex hull of some subset of E containing at most n-\-l points [l, 2 ] . 1 In these theorems the set E is an arbitrary set in the space. The convex hull of E, denoted by H(E), is the set product of all convex sets in the space which contain E. A euclidea...